1. Field of the Invention
The present invention relates to a musical tone synthesizing apparatus which is suitable to synthesize musical tones including anharmonic overtones whose frequencies are not true harmonics of the fundamental frequency.
2. Prior Art
The conventional musical tone synthesizing apparatus, as shown in FIG. 1, has a closed-loop configuration including an adder 1, a delay circuit 2 and a filter 3, all of which are designed as digital circuits. Herein, the delay circuit 2 is constructed by shift registers each further constructed by flip-flops of which number corresponds to the bit number of digital signal supplied from the adder 1. In addition, the clock is supplied to each flip-flop in the shift register by the predetermined sampling period ts. Therefore, delay circuit 2 has delay time tp equal to "Nts" which is obtained by multiplying the sampling period ts by stage number N of shift registers. The filter 3 is designed to apply the predetermined decay characteristic to the signal which propagates through the closed-loop shown in FIG. 1. Herein, transmission-frequency characteristic is adjusted in such a manner that the closed-loop gain becomes slightly smaller than "1".
Herein, the analog signal containing a great number of different frequency components such as the impulse signal is subject to the Pulse-Code Modulation (PCM) by every sampling period ts so that the analog signal is converted into the time-series digital signal, which is to be applied to the above-mentioned conventional musical tone synthesizing apparatus. Such digital signal is applied to the adder 1 and then circulating through the closed-loop consisting of the adder 1, delay circuit 2 and filter 3.
If the phase delay of the filter 3 can be neglected, circulating time of the digital signal which circulates the closed-loop once can be represented by the delay time tp of the delay circuit 2. In this case, the gain-frequency characteristic of this closed-loop has the maximal values at frequencies integral times the fundamental frequency f.sub.1 =1/tp. Since the closed-loop gain is slightly smaller than "1", the signal circulating the closed-loop is gradually attenuated. Then, by effecting the digital-to-analog (D/A) conversion on the output signal of adder 1, it is possible to obtain the musical tone signal containing the fundamental wave and other higher harmonic waves which are produced at frequencies integral times the fundamental frequency f.sub.1. Herein, the amplitude of the musical tone signal is gradually attenuated in lapse of time.
However, the above-mentioned conventional apparatus is disadvantageous in that the delay time tp required for circulating the digital signal through the closed-loop once cannot be set at arbitrary delay time other than delay times integral times the sampling period ts. In order to obtain the delay time shifted from such delay times integral times the sampling period ts, an all-pass filter (APF) 4 is inserted between the delay circuit 2 and filter 3 as shown in FIG. 2. This APF 4 is designed as the primary-stage all-pass filter which is constructed by adders 41, 42, multipliers 43, 44 and a delay circuit 45. In FIG. 2, the delay circuit 2 is constructed by the flip-flops of which number corresponds to the bit number of the digital signal to be transmitting through the delay circuit 2. As similar to the foregoing delay circuit 2 shown in FIG. 1, the clock is supplied to each of the flip-flops in the delay circuit 2 shown in FIG. 2 by every predetermined sampling period ts.
In the APF 4, the adder 41 adds the output of delay circuit 2 to the output of multiplier 44. The output of adder 41 is supplied to the adder 42 via the delay circuit 45, while the delayed signal outputted from the delay circuit 45 is multiplied by multiplication coefficient "-a" and then fed back to the adder 41. In addition, the output of adder 41 is multiplied by multiplication coefficient "a" in the multiplier 43 and then fed to the adder 42. Herein, desirable values in a range between "-1" and "+1" are used as the coefficients "a", "-a". The adder 42 adds the outputs of the delay circuit 45 and multiplier 43 together, and then the addition result thereof is supplied to the filter 3.
Hereinafter., description will be given with respect to the characteristic cf APF 4. In this case, transmission function H(z) of the APF 4 can be represented by the following formula (1). EQU H(z)=(a+z.sup.-1)/(1+az.sup.-1) (1)
As known well, frequency characteristic F(.omega.) can be represented by the following formula (2) by replacing "z.sup.-1 " by exp(-j.omega.ts) in formula (1), wherein ".omega." designates the angular frequency (i.e., .omega.=2.pi.f, f designates frequency). EQU F(.omega.)=[a+exp(-j.omega.ts)]/[1+a exp(-j.omega.ts)] (2)
Next, gain-frequency characteristic G(.omega.) can be represented by the following formula (3). ##EQU1## As indicated in the above formula (3), it can be said that the gain of APF 4 is at the constant value "1" at all frequencies.
Next, phase delay P(.omega.) of the APF 4 can be represented by the following formula (4), wherein arg[F(.omega.)] represents the phase angle of complex function F(.omega.). ##EQU2## By use of approximate calculation tan.sup.-1 (X).apprxeq.X which is used when X is small enough, the above formula (4) can be approximately rewritten to the following formula (5). EQU P(.omega.).apprxeq.sin(.omega.ts)/[a+cos(.omega.ts)]-asin(.omega.ts)/[1+a cos(.omega.ts)] (5)
In the case where the angular frequency ".omega." is very small as comparing to Nyquist angular frequency .omega.n=2.pi.fs/2 and the phase angle .omega.ts is close to zero, approximations such as sin(.omega.ts).apprxeq..omega.ts and cos(.omega.ts).apprxeq.1 can be applied to the above formula (5). Then, the following formula (6) can be obtained. EQU P(.omega.).apprxeq.(1-a)/1+a).omega.ts (6)
Thus, equivalent delay time ta of the APF 4 can be represented by the following formula (7). EQU ta=P(.omega.)/.omega..apprxeq.(1-a)/(1+a)ts (7)
In short, it is possible to adjust the delay time of APF 4 by adjusting the coefficient a. Incidentally, the above-mentioned characteristic of the all-pass filter is described in the paper entitled "Extension of the Karplus-Strong Plucked-String algorithm" written in pages 56 to 69 of the Computer Music Journal, vol. 7, No. 2, 1983 in detail.
Thereafter, it is possible to obtain the resonance characteristic corresponding to the total delay time t=tp+ta in the closed-loop. Next, description will be given with respect to the resonance characteristic of the closed-loop shown in FIG. 2 by referring to graphs shown in FIGS. 3A to 3C.
FIG. 3A shows the relation between the frequency f and phase delay .theta. in the delay circuit 2. As shown in FIG. 3A, when frequency f of the signal passing through the delay circuit 2 is at f.sub.1 =1/tp, the phase difference .theta. is at 2.pi.. Similarly, the phase difference .theta. is at 4.pi. when f is at f.sub.2 which is two times larger than f.sub.1 ; and .theta. is at 6.pi. when f is at f.sub.3 which is three times larger than f.sub.1. In short, the phase delay .theta. increases linearly as the frequency f increases (see line A in FIG. 3A). In addition, when the frequency f is at frequencies integral times the fundamental frequency f.sub.1, both of the input and output signals of the delay circuit 2 are at the same phase.
FIG. 3B shows the relation between the phase delay .theta. and frequency f in the APF 4. As indicated in the foregoing formula (6), while the frequency f belongs to the range whose frequency is very small as comparing to the Nyquist frequency 1/(2ts), the phase delay .theta. varies linearly in proportional to the frequency f. However, if the frequency f is varied in the relatively wide frequency range in the vicinity of Nyquist frequency 1/(2ts), the phase delay .theta. must be varied nonlinearly in accordance with curve B shown in FIG. 3B.
The musical tone synthesizing apparatus as shown in FIG. 2 operates in response to the total phase delay of closed-loop which is obtained by adding the phase delays due to the delay circuit 2 and APF 4 (see FIGS. 3A, 3B). The solid line C in FIG. 3C indicates the total phase delay of closed loop. Therefore, the phase delay .theta. of the digital signal which circulates the closed-loop is turned to be at 2.pi., 4.pi., 6.pi. at frequencies f.sub.1a, f.sub.2a, f.sub.3a which are slightly shifted from frequencies f.sub.1, f.sub.2, f.sub.3 respectively due to the APF 4 to be inserted between the delay circuit 2 and filter 3. When the frequency f is at f.sub.1a, f.sub.2a, f.sub.3a etc., the signal phase is not changed even if the signal circulates the closed-loop so that the closed-loop gain becomes maximal, which indicates the resonance state.
Since the non-linear relation is established between the frequency f and phase delay .theta., the frequencies f.sub.1a, f.sub.2a, f.sub.3a are not disposed at equal intervals. Due to the APF 4, it is possible to synthesize a musical tone containing "anharmonic overtones" whose frequencies are slightly shifted from frequencies integral times the fundamental frequency. In general, "overtones" are defined as harmonic tones whose frequencies are equal to frequencies integral times the fundamental frequency of the note being played. Herein, "anharmonic overtones" are defined as almost harmonic but nonharmonic tones whose frequencies are slightly shifted from frequencies integral times the fundamental frequency (see U.S. Pat. No. 3,888,153). By use of the filter in which the frequency varies non-linearly with respect to the phase delay, it is possible to synthesize the musical tone containing the anharmonic overtones, which is disclosed in U.S. Pat. No. 4,130,043.
However, the musical tone actually sounded from the nonelectronic musical instrument (i.e., acoustic instrument) has the anharmonic overtones whose frequencies are quite shifted from frequencies integral times the fundamental frequency. Particularly, in case of the percussion instrument, its percussion tone to be sounded contains the anharmonic overtones whose frequencies are quite different from frequencies integral times the fundamental frequency. However, the conventional musical tone synthesizing apparatuses described herein cannot produce the anharmonic overtones whose frequencies are quite shifted from frequencies integral times the fundamental frequency. Thus, there is a problem in that the conventional apparatus cannot synthesize the musical tone having the high-fidelity to the harmonic and anharmonic overtone structure of the sound of acoustic instrument such as the percussion instrument.